Concentric 3x3 magic square

 

What, a concentric 3x3 magic square? But you told us, there is only 1 (x 8 because of rotation and/or mirroring) simple 3x3 magic square?

 

The 3x3 magic square is more than simple. It is symmetric and yes it is also concentric, because the middle number is in the 1x1 centre and the lowest and highest numbers are in the border. In the border the lowest number is opposite to the highest number, the second lowest number is opposite to the second highest number, ...

 

 

4

3

8

9

5

1

2

7

6

 

 

Use the 3x3 concentric magic square to construct a 3x3 in 5x5 concentric magic square. Add 8 to all numbers of the 3x3 magic square and construct the 5x5 border with the lowest and highest numbers. Add 12 to all numbers of the 5x5 concentric magic square and construct the 7x7 border with the lowest and highest numbers and so you get a 3x3 in 5x5 in 7x7 concentric magic square. You can put borders around the concentric magic square to infinitity.

 

Take for example a 3x3 in 5x5 in 7x7 in 9x9 concentric magic square. If you remove the 9x9 border, you get an (impure) 7x7 magic square. Remove the 7x7 border and you get a 5x5 magic square. Remove the 5x5 border and you get a 3x3 magic square.

 

It is also possible to construct concentric magic squares of even order. The centre of the even magic square is a 2x2 square (N.B.: It is not possible to put the 4 middle numbers in the centre and getting a valid magic square, but in the 4x4 inlay of a 4x4 in 6x6 concentric magic square you must put the middle numbers). Construct the border with the lowest and highest numbers again and again and you get a 4x4 in 6x6 in 8x8 in ... concentric magic square.

 

See how the concentric magic square on this website is growing bigger and bigger: 

3x34x45x56x67x78x89x910x1011x1112x1213x1314x1415x1516x1617x1718x1819x1920x2021x2122x2223x2324x2425x2526x2627x2728x2829x2930x3031x31 and 32x32